Further Contents of E = mc2

Let us look at the energy-momentum relations for massive and massless
particles. They are E = p2/2m and E = cp respectively.
As you know, it was Einstein who unified these relations with his
special relativity.

Birth Place of E = mc2

In this house,
during the years 1903-1905,
Albert Einstein created his fundamental treatise on relativity theory.

As for the internal space-time symmetries, massive particles have
the O(3)-like symmetry which produces the concept of spin.
Massless particles have the symmetry which is like the two-dimensional
Euclidean group. Wigner made these observations in his 1939 paper
on "inhomogeneous Lorentz group" where introduces the little group as
the subgroup of the Lorentz group whose transformations leave the
four-momentum of a given particle invariant.

If the massive particle is at rest, its O(3)-like symmetry leads to
the concept of spin. For the massless particle, the rotational
degree of freedom of the E(2)-like little group corresponds to the
helicity. The translational degrees of freedom have been shown to
correspond to gauge transformations. Indeed, massive and massless
particles appear to have two distinct internal space-time symmetries.
Can they be unified as Einstein did for the energy-momentum relation?

In order to answer this question, let us boost a particle with spin.
Its longitudinal component remains invariant and is called the helicity.
How about its transverse components? They become contracted to the
translational-like degrees of freedom in the E(2)-like little group,
which physically are gauge degrees of freedom.
Together with D. Han and D. Son, I published
a series of papers in the 1980s establishing that Wigner's little
group together with group contraction techniques unifies the
internal space-time symmetries of massive and massless particles.

You are then invited to build your own house. How are you going to
build your own house? With what? In order to build a house consistent
with it neighborhood is to find a Lorentz-covariant entity which
takes different forms for slow and fast particles.